Asymptotic Beam Theory and Buckling Analysis Based on Implicit Constitutive Relation


Student thesis: Doctoral Thesis

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Award date25 Jul 2022


An implicit constitutive relation formulation for solid mechanics is proposed as a new sort of paradigm to describe the mechanical behavior of elastic solids by using the left Cauchy-Green deformation tensor B and Cauchy stress T, which contains the classical Cauchy elastic model and Green elastic model. Under the assumption wherein the displacement gradient is sufficiently small, linearization can lead to a nonlinear relation between the infinitesimal strain tensor ε and T. This thesis is devoted to the derivation of an asymptotic beam theory and buckling analysis based on a subclass of implicit constitutive relation.

In the first part, we are devoted to studying the plane-stress deformation of a beam, which is described by a nonlinear constitutive relation between the linearized strain and stress. This relation is obtained from an implicit one by linearizing it under the assumption wherein the displacement gradient is sufficiently small. Moreover, this relation is suitable for modeling certain inter-metallic alloys. Further, the aim is to derive a consistent asymptotic beam theory without the ad hoc assumptions usually made in the development of beam theories. The methodology involves expanding the displacement, in plane-strain tensor, and stress tensor in a Taylor series, leading to a system of nonlinear equations that are then solved. An analytical iteration procedure is developed to solve the system of equations leading to an analytical solution. The beam theory and the approximate general analytical solutions are used to study four examples. For validation of the approximate analytical solution, we use a spectral collocation method to carry out numerical simulations for the full 2D problem, which confirms the validity of the approximate analytical solution. The study also reveals that the Euler-Bernoulli type of hypothesis is not suitable for a certain class of problems.

In the second part, we consider the buckling of elastic solids described by a subclass of implicit constitutive relations. We present a general nonlinear incremental theory, which will provide a theoretical basis to figure out the post-buckling behavior of the elastic bodies described by a subclass of implicit constitutive relation. Besides, by linearizing the nonlinear incremental theory, we carry out a bifurcation analysis of a uniaxially compressed rectangular layer described by an implicit constitutive relation. We then provide general governing equations regarding the mixed unknowns, i.e., displacement and stress fields, within the framework of finite strain deformation. Thereby, the combination of the asymptotic numerical method (ANM) and spectral collocation method is applied to solve the resulting nonlinear equations. We validate our numerical framework by comparing the computational results with analytical ones and then explore the effects of width-to-length ratio and material nonlinearity on the buckling and post-buckling behavior. The larger the width-to-length ratio is, the larger the critical buckling load is. The material parameter, η, has a significant impact of up to 28% on the critical buckling load and influences up to 50% on the post-buckling behavior. Our model based on the implicit constitutive relation well predicts the buckling and post-buckling of Gum metal alloy. In the last part, a conclusion including a summary is given.